How many unique ways are there to arrange the letters in the word APE?
Solution: Let's try building the re-arrangements (or permutations) letter by letter. The word is $3$ letters long: _ _ _ For the first blank, we have $3$ choices of letters. After we put in the first letter, let's say it's $\text{P}$, we have $2$ blanks left. $\text{P}$ _ _ For the second blank, we only have $2$ choices of letters left. So far, there are $3 \cdot 2$ unique choices we can make. We can continue in this fashion to put in a third letter, and so on. At each step, we have one fewer unique choice to make, until we get to the last letter, and there's only one we can put in. So, the total number of unique re-arrangements must be $3\cdot2\cdot1$. Another way of writing this is $3!$, or $3$ factorial, which is $6$.